The Java files are installed in a directory called jre1.7.0_07 in the current directory.
In this example, it is installed in the /usr/java/jre1.7.0_07 directory.
When the installation has completed, you will see the word Done.
Delete the .tar.gz file if you want to save disk space.
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My question is:

Once jre-7u7-linux-i586.tar.gz has been uncompressed, what else needs to be done?
I did not see the word "Done".
The program is uncompressed but not installed at that point, correct?
The instructions do not say what command should be entered to run the installer.
In my case would that be:
sudo /usr/java/jre1.7.0_13/bin/java
???

"You only need to uninstall it to start it using it. You could now proceed to add it to your path by updating your profile file or make it the default if you had others by running"

1. I was not talking about uninstalling anything.
2. You only need to uninstall it to start it using it"??? What does that mean? Did you mean "install"?
3. What must I add to myu path? PATH=$PATH:/usr/java/jre1.7.0_13/bin ???
4. What profile should I update? I updated the $PATH in ~/.bashrc
5. What was this command supposed to do? sudo update-alternatives --config java
6. If I go to Ubuntu Software Center -> Installed Software. I do not see that Java has been installed.

3. What must I add to myu path? PATH=$PATH:/usr/java/jre1.7.0_13/bin ???

Yes - basically just that but I do a couple of extra things. I make a soft link in /usr/java from /usr/java/jdk to /usr/java/jre1.7.0_13 and then modify the PATH at the end of /etc/bash.bashrc using PATH=$PATH:/usr/java/jre/bin .

This means that all profiles will now be able to use the JRE and I can have multiple JRE version installed in /usr/java and just change the soft link to change which JRE is being used.

6. If I go to Ubuntu Software Center -> Installed Software. I do not see that Java has been installed.

You don't need to see it installed. It is in effect installed once you update the PATH.

Since you are only installing the JRE and not the JDK I assume you only want to run already compiled Java programs and not actually program in Java yourself. Is that right?

Oh dear, I meant extract. You only need to extract it. Sorry for the typo.

3.) Something like that, yes
4.) exact file depends on your build. This is all optional ofcourse. All we are trying to do is set a default which is not strictly required
5.) Locates all the java programs available on your computer and asks you to chose which one you want to set as default
6.) I don't know I've never used that software center